1a) for an organic dairy cow, shortly after calving, assume that the amount of milk produced follows a normal distribution with a mean of 27 litres of milk per day and a standard deviation of 4 litres of milk per day.
i) calculate the probability that an organic dairy cow will produce less than 33 litres of milk per day.
My answer - 0.9332
ii) calculate the probability that an organic dairy cow will produce between 24 litres and 33 litres of milk per day.
My answer - 0.7066
B) later in the milking cycle the amount of milk produced still follows normal distribution. However, the mean value will decline but the standard deviation will remain the same. Calculate the new mean value for the amount of milk produced each day at this point in the milking cycle if the probability that an organic diary cow will produce more than 16 litres of milk per day is 0.98.
I don't understand how to part B. Please explain
Thankyou
1st parts ok
Using a standard Z table, 98% of the area is above the -2.054 std. So, the new mean is
16 + 4*2.045 = 24.18
To solve part B, we need to find the new mean value for the amount of milk produced each day at this point in the milking cycle. We are given that the probability of producing more than 16 litres of milk per day is 0.98.
To do this, we can use the standard normal distribution table, also known as the z-table. The z-table gives the cumulative probability of values under the standard normal distribution.
Step 1: Find the z-score corresponding to a cumulative probability of 0.98.
We want to find the z-score that corresponds to a cumulative probability of 0.98 (which represents the area under the curve to the left of the z-score). By looking up this probability in the z-table, we find that the z-score is approximately 2.05.
Step 2: Use the z-score formula to find the new mean value.
The z-score formula is: z = (x - mean) / standard deviation
Rearranging the formula, we get: x = z * standard deviation + mean
We know the z-score (2.05), and we know the standard deviation (4 litres per day). We want to find the new mean (x) when the lower limit is 16 litres of milk per day. So we can plug these values into the formula:
16 = 2.05 * 4 + x
Simplifying the equation:
16 = 8.2 + x
Subtracting 8.2 from both sides:
x = 7.8
Therefore, the new mean value for the amount of milk produced each day at this point in the milking cycle is approximately 7.8 litres.